A recent update on the existence of extremal Kaehler metrics in Kaehler surfaces

Abstract: In this talk, we will first give a brief ``tour" of Kaehler geometry and discuss some key results obtained in recent years. For instance, the uniqueness of extremal Kaehler metric as well as the existence of a sharp lower bound of the Calabi energy in any Kaehler class. We then discuss the existence of extremal Kaehler metrics in Fano surface via continuous deformation method. In particular, we prove the existence of a new Einstein metric in CP2 \sharp 2 \bar{CP}2 with positive scalar curvature. Combining with famous theorem of G. Tian, this gives a final and affirmative answer to a long standing question: does every Fano surface admits an Einstein metric with positive scalar curvature?

Refreshments will be served at 2:45 p.m. (PIMS Lounge).